Path Analysis for Recursive Generalized Linear Model Systems
Nobuoki Eshima1, Minoru Tabata2, Tetsuji Ohyama1
1Biostatistics, Oita University, Yufu, Oita, Japan; 2Mathematical Sciences, Oita Prefecture University, Sakai, Osaka, Japan

Path analysis is usually carried out in causal systems of continuous variables, i.e. Linear Structural Equation Model (LISREL). In LISREL approach, causal relationships among variables concerned are translated into linear equations of the variables, and causal effects are calculated with regression coefficients and correlation coefficients. In comparison with path analysis of continuous variables, that of categorical variables is complex. Hagenaars (1998) made a discussion of path analysis of categorical variables by using a loglinear model approach. Although the approach is an analogy to LISREL, the discussion of the direct and indirect effects was not made. Eshima et al. (2001) proposed a method of path analysis of categorical variables by using logit models. In this approach, the direct and indirect effects of variables are discussed according to log odds ratios. Kuha & Goldthorpe (2010) also proposed a similar path analysis method for categorical variables; however increasing categories in variables makes the path analysis to be complex. In practical data analyses, there are many cases of non-normal response variables in various fields of studies. In order to assess the effects of factors or explanatory variables in generalized linea models (GLMs) it may be appropriate to use one of predictive power measures for this objective. The entropy correlation coefficient (ECC) and the entropy coefficient of determination (ECD) were proposed as predictive power measures for GLMs (Eshima & Tabata, 2007, 2010), and the measures are extensions of the multiple correlation coefficient and the coefficient of deremination in the ordinary linear regression model, respectively.

This paper proposes a basic method for path analysis in causal systems with GLMs.The total, direct and indirect effects in GLMs are discussed by using ECC and ECD, and a path analysis method of recursive GLM systems is proposed. Numerical examples are also given to illustrate the present approach.


Eshima, N. & Tabata, M. (1999). Effect analysis in loglinear model approach to path analysis of categorical variables, Behaviormetrika; 26: 221-233.

Eshima, N. & Tabata, M. (2007). Entropy correlation coefficient for measuring predictive power of generalized linear models, Statistics and Probability Letters; 77, 588-593.

Eshima, N & Tabata, M (2010) Entropy coefficient od determination for generalized linear models, Computational Statistics and Data Analysis, 54, 1381-1389.

Eshima, N., Tabata, M. & Geng, Z. (2001). Path analysis with logistic regression models: effect analysis of fully recursive causal systems of categorical variables, Journal of the Japan Statistical Society; 31: 1-14.

Hagenaars, J. A. (1998). Categorical causal modeling: latent class analysis and directed loglinear models with latent variables, Sociological Methods & Research; 26: 436-489.

Kuha, J. & Goldthorpe, J. H. (2010) Path analysis for discrete variables: the role of education in social mobidity, J. R. Statist. Soc. A, 1-19.

Keywords: Direct Effect; Indirect Effect; Entropy Correlation Coefficient; Generalized Linear Model

Biography: Nobuoki Eshima was born in Fukuoka, Japan in 1957. He was received B. Sc. and D. Sc. degrees in Mathematics from Kyushu University, Fukuoka, Japan in 1980 and 1993, respectively. In 1993, he joined Department of Statistics, Faculty of General Education, Nagasaki University as Associate Professor. In 1996, he joined Department of Biostatistics, Faculty of Medicine, Oita Medical University as Professor. Since 2003 he is working as a Professor in Faculty of Medicine, Oita University.